具有完全退化半稳定约化的曲线的下降映射

p-adic域上全退化的algebraic varieties的余上同调

On the etale cohomology of algebraic varieties with totally degenerate reduction over p-adic fields

设K是Q_p的有限扩张，X是K上的光滑射影簇.定义了K的整数环上X的正则真模型的特殊纤维的全退化约化和相应的Chow复形。若X有这样的约化，则证明了对任意l，X的Q_l-进余同调有一个滤子，其分次积作为Galois模同构于有限维Q-向量空间（具有有限的Galois非分歧作用）的张量积，且Q_l的扭具有分圆特征标.

Newton 非退化函数的邻近周期公式下降

Descent of nearby cycle formula for Newton non-degenerate functions

本文证明 Newton 非退化函数邻近周期公式及其 motive 版本的下降定理，无需便利条件，并由此推出 Milnor 单值性最大 Jordan 块数公式及单纯形情形下修正的 Steenbrink 猜想。

Ramification and descent in homotopy theory and derived algebraic geometry

We introduce notions of unramified and totally ramified maps in great generality - for commutative rings, schemes, ring spectra, or derived schemes. We prove that the definition is equivalent to the classical definition in the case of rings of integers in number fields. The new definition leads directly (without computational techniques) to a calculation of topological Hochschild homology for rings of integers. We show that THH(R) is the homotopy cofiber of a map $R[ΩS^3]\otimes_RΩ^1_{R/\mathbb{Z}}\to R[ΩS^3\langle 3\rangle]$, so there is a long exact sequence $\cdots\to H_\ast(ΩS^3;Ω^1_{R/\ma…

许多奇异点并不稳定地等价于牛顿非退化奇异点

Many singularities are not stably equivalent to Newton-non-degenerate singularities

这篇论文被撤回了。考虑一个孤立的复超曲面奇点f（x_1，..，x_n）=0。对于牛顿非退化奇点，局部拓扑完全由相关的多面体对象（牛顿图）确定。对于任何局部坐标的选择，“大多数”奇点都不是牛顿非退化的。Arnol'd的一个老问题是，对于任何超曲面奇点，是否存在稳定性f（x_1，...，x_n）+z^2_1+.+ z^2_r，在坐标发生一些变化后，它变成牛顿非简并的。答案是：“完全没有”。我们给出了一些简单的障碍，并提出了特殊的例子平面曲线奇点，没有牛顿非退化稳定（在任何局部坐标）。

A combinatorial description of the dormant Miura transformation

A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat $\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant generic Miura $\mathfrak{sl}_2$-opers on a totally degenerate curve. The combinatorial objects that we use are certain branch numberings of $3$-regular graphs. Our description may be thought of as an analogue of the combinatorial description of dormant $\mathfrak{sl}_2$-opers given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the Miura tra…

Trigonal Morsifications on Hirzebruch Surfaces with an appendix by E. Shustin

In this paper we obtain a classification of rigid isotopy classes of totally reducible trigonal curves lying on a Hirzebruch surface $Σ_n$, and having a maximal number of non-degenerated double points. Such curves correspond to morsifications of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ (the union of three smooth real branches intersecting each other with multiplicity $n$). We obtain this classification by studying combinatorial properties of dessins. In the appendix, we prove that any morsification of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ …

Galois descent of semi-affinoid spaces

We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a $K$-analytic space $X$, provided that $X\otimes_KL$ is semi-affinoid for some finite tamely ramified extension $L$ of $K$. As an application, we study the forms of analytic annuli that are trivi…

正交Grassmannian的不可压缩性

Incompressibility of orthogonal grassmannians

我们证明了Bryant Mathews（2008）的猜想。设Q是任意域上非退化二次型q的全迷向i-平面的正交格拉斯曼算子（其中i是区间[1，（\dim q）/2]中的整数）.如果Q上每个闭点的度都能被2 ^i整除，且q在Q的函数域上的Witt指数等于i，则簇Q是2-不可压缩的。

p-adic Tate conjectures and abeloid varieties

We explore Tate-type conjectures over $p$-adic fields. We study a conjecture of Raskind that predicts the surjectivity of $$ ({\rm NS}(X_{\bar{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_K} \longrightarrow H^2_{\rm et}(X_{\bar{K}},\mathbb{Q}_p(1))^{G_K} $$ if $X$ is smooth and projective over a $p$-adic field $K$ and has totally degenerate reduction. Sometimes, this is related to $p$-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether $$ {\rm Hom}(A,B)\otimes{\mathbb{Q}}_p \,\to\, {\rm Hom}_{G_K}(V_p(A),V_p(B)) $$ is surjective if $A$ and $B$ ar…

半阿贝尔Néron模型的相对紧化，II

Relative compactification of semiabelian Néron models, II

本文研究半阿贝尔Néron模型的相对紧化问题，证明了满足特定Cohen-Macaulay性与层条件的唯一相对紧化的存在性，先后讨论部分退化情形及基环为Dedekind整环的情形，补充了此前完全退化情形的相关研究。